Self-similar solutions of energy-supercritical focusing wave equations in all dimensions

Abstract

In this paper, we prove the existence of a countable family of regular spherically symmetric self-similar solutions to focusing energy super-critical semi-linear wave equations \begin{equation*} \partial_{tt}u-Δu=|u|^{p-1}u \qquad \text{in} \,\, \mathbb{R}^{N}, \end{equation*} where $N\ge 3$, $1+\frac{4}{N-2}<p$, and, if $N\ge 4$, $p\le 1+\frac{4}{N-3}$. This was previously known only in the case $N=3$, for integer $p$ (see Bizoń, Maison and Wasserman \cite{BMW}). We also study the asymptotics of these solutions.